 So we've been talking about the Buckingham Pi method as a way to be able to connect experiments efficiently. What we're going to do now is take a look at the six steps involved in determining the Pi parameters. It will be a little on the abstract side but bear with me and then in the next segment we'll apply it to an example problem. So if you recall what we've been working on is if you have some dependent and then a bunch of independent variables up to Qn, what we're trying to do is recast this as a system with Pi parameters and Pi are non-dimensional variables and what we've been saying is that we can take those n variables and recast them as n minus m non-dimensional variables where m is the number of primary dimensions required to specify Q1 through Qn. So what we're going to do now we're going to come up with the step-by-step procedure and being able to go from this to these Pi parameters. We're not going to get the functional relationship and in order to get that you have to do experiments. So let's take a look at the step-by-step procedure. So the first step is to list all the parameters involved. When we looked at drag on a sphere we talked about the force, we talked about the velocity, we talked about the diameter, we talked about the density, the viscosity. Those are the main parameters involved and in this case we would have 1, 2, 3, 4, 5. So n would be 5 when we looked at the sphere. Second step, second step is to select a set of fundamental or primary dimensions. So example, if you're dealing with mass, length, time, we saw that earlier in the course. If you have non-isothermal flow, for example if you're dealing with a problem in heat transfer where you're trying to come up with some non-dimensional numbers there, with that you would have mass, length, time, and theta. Remember we use theta for temperature. So it all depends upon the nature of the problem but typically in fluid mechanics for incompressible flow and non-isothermal flow we're dealing with MLT, so mass, length, time. So that's the second thing. Third thing would be to list all of your parameters in terms of the primary dimensions and r would be then the number of primary dimensions. So that would be mass, length, time. Typically r is 3 but sometimes, like I said, it could be 4. The step 4 is to go back to list number 1, so go back to step 1. And what you want to do is you want to select r parameters which collectively contain all of the primary dimensions. And so that means for if you have 3 primary dimensions you want to take 3 of these, so maybe velocity, diameter, and density. And those will form kind of a nucleus group that we're then going to work with and add the other parameters to those 3. And I picked vd-roll kind of by cheating but that's quite often what we will use in fluid mechanics. And because they themselves represent all the primary dimensions and they are not scaling of one another. For example, if you have diameter and diameter squared, that would be length and length squared. You wouldn't want to do something like that. You would want to ensure that you're getting different mixes of your primary dimensions when you're selecting these r parameters. And the other thing that you want to do is you want to make sure that the dependent variable is not one of them. So what does that mean? If you're looking at using the example that we looked at earlier, with the drag on a sphere, you would want to ensure that the force, that is the drag force, is not one of the three that you're using to come up with this r parameters for step 4. So you want to ensure that force is not going to be one of those because that is your dependent variable. You want that not to be included. So that is step 4. Let's move on and take a look at step 5. So step 4 is you want to set up dimensional equations. And from an earlier segment, you remember we talked about dimensional homogeneity. Essentially what we're going to be doing is setting the dimensions of our pi parameters equal to 0. And then we get a number of equations, a number of unknowns. And this, in all honesty, is probably the funnest part of Buckingham Pi because you get to solve these equations and then you get your exponents. And I'll show you in the example in the next segment what I'm talking about. But here, another thing that can sometimes be confusing, we had r being the primary dimensions, and then I have this m. What the m is referring to would be m is equal to 3 in this particular example. So it refers to the cluster of variables that you pull out for this part of step 4. So if you're wondering what m is, that is what m is referring to. So we set these equations up and what we're going to get out of this, we're going to get our pi 1, pi 2, all the way up to pi n minus m. And those should all be non-dimensional and they will consist of the different variables that are important for your experiment. And the final step that we need to do is do a sanity check, and so we check the dimensions of our resulting pi parameters. So if you've done your work correctly, this should happen automatically, but you always want to do a sanity check to make sure you didn't make a mistake somewhere. So that is the technique or the step-by-step procedure for the Buckingham Pi analysis. What we're going to do in the next segment is we're going to work a couple of example problems to see this in action and it comes up with our pi parameters that you can then use to scale your experiments with. So that is the technique for Buckingham Pi.